The M\"obius function

The Möbius function $\mu (n)$ is defined as if is a product of distinct primes, and as if the square of a prime divides . Define :

$\begin{displaymath} M(x) := \sum_{n \leq x} \mu (n). \end{displaymath}$

Then RH is equivalent to each of the following statements

$\begin{displaymath} M (x) \ll x^{1/2 + \epsilon}, \end{displaymath}$

for every positive $\epsilon$ ;

$\begin{displaymath} M (x) \ll x^{1/2} \exp (A \log x/\log\log x); \end{displaymath}$

for some positive A.

Both results are due to Littlewood.

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